Rewrite as a simplified fraction. ${2.2\overline{1} = {?}}$
Solution: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}100x &= 221.1111...\\ 10x &= 22.1111...\end{align*}}$ Subtracting the two equations, we have: ${90x = 199}$ ${x = \dfrac{199}{90}} \text{or} {2 \dfrac{19}{90}}$